1896.J Harmonics of unrestricted Degree, Order, fyc. 191 



where mod. ( 1 ~ u \ < 1 ; the upper or lower signn is to be taken in the 

 I 2 /_ 



exponential, according as the imaginary part of ju is positive or 

 negative. Degenerate forms of these series for special restrictions as 

 to n and m are considered. 



The following expressions for P,/"(u), Q u m (,n) when mod. /u>l are 

 obtained — 



P m (/ A - »fr n(n + m) _ ( b n _ m . x 



/?z + m + 2 rc+m + l 3 l\ lftJ n(«— I) 



n(»+i) 



e »«ri n(» + m)n(-4) , „ N . 



„ /n + ra-f 2 w + m + l 3 1 

 F , , n-\ — , — 



\ 2 2 2 j» s 



The following relations between the particular integrals of the 

 differential equation obtained by changing n into — n— 1, and m into 

 — m, are found ; by means of these relations the eight solutions are 

 expressed in terms of two of them — 



g — #nri 



Pwf "W = v cos ?Mr {Q» W W sinC* + m)--Q_:-i(,") sin (n-ni)ar}, 

 P^'O) =3 -) ( {P« ot O) — e-^sinwjr . Q/<0)}. 



11^72. -\-t7l) 7T 



It is further shown that 



Q»"(-aO = -e±^Qn m (ri, 



where the upper or lower sign in the exponentials is taken according 

 as the imaginary part of f.i is positive or negative. 



The following expressions for the functions are obtained for the 

 domain of fx — : — 



